If, for example, you have a variety show, and a guest sings a song, then that song is likely copyrighted.

Now, another member on this site has gotten in writing that BMI is the only agency that requires Part 15 broadcast licensing (and as far as I know, it's a blanket agreement, so much per year). ASCAP & SESAC do not require licensing. You'd be safe in playing ANY OTR show, even those containing copyrighted songs, if you obtained that BMI Part 15 blanket license.

If, however, the show is a play, for example, or a comedy, with incidental music only, then it could be considered to be in the public domain. Many of the older shows, such as Lights Out, Lux Radio Theater, etc., are generally agreed to be in the public domain. Each, though, has to be taken on a case by case basis.

Sad, but true.

As another example, I read a while ago that that the early Lum & Abner shows had been ruled to be in the public domain (when a purported rights holder came out of the woodwork), while the later ones were not.

As for old music, songs written prior to 1922 are considered to be in the public domain in the U.S. (other countries have different rules). But again, if you obtain that Part 15 blanket license, you can play (over-the-air, not streaming) any music you want. Streaming requires a different set of licenses that are even more complicated.